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June  2021, 41(6): 2947-2969. doi: 10.3934/dcds.2020392

Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems

Jose Anderson Cardoso 1, , Patricio Cerda 2, , Denilson Pereira 3, and  Pedro Ubilla 2,,

1. 

Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, 49100-000, Brazil
2. 

Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
3. 

Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil

* Corresponding author

Received  December 2019 Revised  October 2020 Published  June 2021 Early access  December 2020

Fund Project: The first author is partially supported by FAPITEC/CAPES and by CNPq - Universal.
The second author was partially supported by Proyecto código 042033CL, Dirección de Investigación, Científica y Tecnológica, DICYT.
The third author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT.
The fourth author was partially supported by FONDECYT grant 1181125, 1161635, 1171691

We prove the existence of a bounded positive solution for the following stationary Schrödinger equation
$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $
where
$ V $
 is a vanishing potential and
$ f $
 has a sublinear growth at the origin (for example if
$ f(x,u) $
 is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [6]. In addition, if
$ f $
 has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type
$ \rho(x)f(u) $
 where
$ f $
 is a concave-convex function and
$ \rho $
 satisfies the
$ \mathrm{(H)} $
 property introduced in [6]. We also note that we do not impose any integrability assumptions on the function
$ \rho $
, which is imposed in most works.
Keywords: Concave-convex nonlinearities, upper and lower solutions, variational methods, Schrödinger equation, bounded solutions.
Mathematics Subject Classification: 35J20, 35J10, 35J91, 35J15, 35B09.
Citation: Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392
References:
[1]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.

[3]

A. Bahrouni, H. Ounaies and V. D. Rădulescu, Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.  doi: 10.1007/s13398-018-0541-9.

[4]

A. Bahrouni, H. Ounaies and V. D. Rădulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.

[5]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.

[6]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[8]

J. Chabrowski and J. M. B. do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.

[9]

D. G. de Figueiredo, J-P Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.

[10]

D. G. de Figueiredo, J-P Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.  doi: 10.4171/JEMS/52.

[11]

F. Gazzola and A. Malchiodi, Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.  doi: 10.1081/PDE-120002875.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.

[13]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997.

[14]

T-S Hsu and H-L Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.  doi: 10.1016/j.jmaa.2009.12.004.

[15]

Z. Liu and Z-Q Wang, Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.

[16]

M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967.

[17]

T-F Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.

show all references

References:
[1]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.

[3]

A. Bahrouni, H. Ounaies and V. D. Rădulescu, Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.  doi: 10.1007/s13398-018-0541-9.

[4]

A. Bahrouni, H. Ounaies and V. D. Rădulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.

[5]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.

[6]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[8]

J. Chabrowski and J. M. B. do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.

[9]

D. G. de Figueiredo, J-P Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.

[10]

D. G. de Figueiredo, J-P Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.  doi: 10.4171/JEMS/52.

[11]

F. Gazzola and A. Malchiodi, Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.  doi: 10.1081/PDE-120002875.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.

[13]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997.

[14]

T-S Hsu and H-L Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.  doi: 10.1016/j.jmaa.2009.12.004.

[15]

Z. Liu and Z-Q Wang, Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.

[16]

M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967.

[17]

T-F Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.

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